CE5504 – Surface Water Quality Modeling

CE5504 – Surface Water Quality Modeling

CE5504 Surface Water Quality Modeling

Lab 1. Modeling 101

dcVdt

dcVdt

discoverydiscovery applicationapplication

fate andtransport

decisionsupport

integrationintegration

Surface Water QualityEngineering

Reactor Analogs

CE5504 – Surface Water Quality Modeling

Plug FlowReactor(rivers)

Completely-MixedFlow Reactor(lakes, bays, nearshore)

Mille Lacs Lake, Minnesota

Fox River, Wisconsin

The Reactor Analog for Lakes

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

Completely Mixed Flow Reactor

CMFR

The Mass Balance

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

CMF Reactor Characteristics

• completely mixed (Cout = C)

• constant volume (Qin = Qout)

The Mass Balance

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

Control Volume

• the system about which the mass balance will be computed.

The Mass Balance

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

Kinetics

• growth• decay RXN

RXN

The Mass Balance

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

Kinetics

• 0 order reactions

rate is not a function of concentration

Dollar Bay

0

2

4

6

8

10

12

0 30 60 90 120 150 180 210 240 270 300 330 360Day of Year

Dis

solv

ed O

xyge

n (m

g/L)

Dollar Bay

y = -0.1285x + 23.752R2 = 0.9759

0

2

4

6

8

10

0 30 60 90 120 150 180 210 240 270 300 330 360Day of Year

Dis

solv

ed O

xyge

n (m

g/L)

Zero Orderk = 0.13 mg∙L-1∙d-1

Ct = C0 - k∙t

k, mg·L-1·d-1

The Mass Balance

CE5504 – Surface Water Quality Modeling

( )0

ttX X e

Kinetics

• 1st order reactions

rate is a function of concentration

lnCt = -k∙t + lnC0 or

Ct = C0·e-k·t

k, d-1

Pb-210

y = -0.036x + 6E-16R2 = 1

-4

-3

-2

-1

0

0 20 40 60 80 100Time (yr)

Rad

iois

otop

e C

once

ntra

tion

Pb-210

0.0

0.2

0.4

0.6

0.8

1.0

0 10 20 30 40 50 60 70 80 90 100Time (yr)

Rad

iois

otop

e C

once

ntra

tion

Writing the Mass Balance

CE5504 – Surface Water Quality Modeling

RXNin

dCV Q C Q C V k Cdt

3 33 3

3 3 3 3

1g m g m g gm mm d d m d m d m

1st orderdecay

At Steady State

CE5504 – Surface Water Quality Modeling

RXN1st orderdecay

inQ C V k C Q C

indCV Q C Q C V k Cdt

0

( ) inC Q V k Q C

ss inQC C

Q V k

At steady state, the source terms are equal to the sink terms and there is no net change in mass within the control volume.

Time Variable(analytical solution)

CE5504 – Surface Water Quality Modeling

indCV Q C Q C V k Cdt

,1 ,1ss inQC C

Q V k

,2 ,2ss inQC C

Q V k

conc

entra

tion

time

Css,1

Css,2

,1 ,2 1Q Qk t k tV V

t ss ssC C e C e

flushing out building in

Time to Steady State Variable

CE5504 – Surface Water Quality Modeling

conc

entra

tion

time

Css,1

Css,2

ln (1 )1tk

95

ln 0.051

tk

,1 ,2 1Q Qk t k tV V

t ss ssC C e C e

Noting that the hydraulic retention time, = V/Q

1 1

,1 ,2 1k t k t

t ss ssC C e C e

and (Chapra, Sec. 3.3)

or, for 95% or 953

1tk

Variability in and k

CE5504 – Surface Water Quality Modeling

953

1tk

Lake (years)Superior 179

Michigan 136

Ontario 8

Onondaga 0.25

Material k (yr-1)Organic C 36.5

Atrazine 1.0

PCB 0.05

Chloride 0

Review

CE5504 – Surface Water Quality Modeling

1. Can you see any limitations to the analogs in Slide 3?

2. Can you identify 2 additional source terms for lakes in Slide 4?

3. Discuss the completely mixed and constant volume assumptions in Slide 5.

4. Develop an example of an inappropriately-defined control volume in Slide 6.

5. Provide some additional examples of growth and decay in Slide 7.

6. Show how a system acts to bring itself to steady state; see Slide 10.

7. In lab -

a. Determine half-lives of selected chemical species.

b. Compare response times of lake/chemical couplets.

c. Calculation of steady state concentrations.

d. Time variable solutions: kinetics and step function response.